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SEMICONDUCTOR TECHNICAL DATA
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Prepared by: Roy Hejhall
Applications Engineering
This note explains high–frequency transistor response parameters and discusses their interdependance. Useful
nomograms are given for determining hfe, fT, fαe, fmax, and many other parameters.
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ABSTRACT
The range of frequencies over which a transistor performs
a useful circuit function is limited by inherent parameters.
Manufacturers’ data sheets often specify only one or two of
these parameters, and questions concerning others often
arise. Therefore, a clear understanding of these parameters
is of value in attempting to answer such questions from the
data given.
PARAMETER CHARACTERISTICS AND
INTERRELATIONSHIPS
One parameter is hfb (alpha, the common base ac
short–circuit forward current gain). As frequency is
increased, hfb remains approximately equal to hfbo (the value
of hfb at 1 kHz). After the upper frequency is reached, hfb
begins to decrease rapidly.
The frequency at which a significant decrease in hfb
occurs provides a basis for comparison of the expected high
frequency performance of different transistors. The common
base current gain cutoff frequency, fαb, is defined as that
frequency at which hfb is 3 dB below hfbo. Expressed in
magnitude, hfb at fαb, is 70.7 percent of hfbo. Power gains,
current gains, and voltage gains for a few common decibel
values are found in Table 1. A curve of hfb versus frequency
for a transistor with an fαb of 1 MHz is shown in Figure 1.
This curve has the following significant characteristics:
(1) at frequencies below fαb, hfb is nearly constant and
approximately equal to hfbo, (2) hfb begins to decrease
significantly in the region of fαb, (3) above fαb, the rate of
decrease in hfb with increasing frequency approaches 6 dB
per octave in the limit.
The curve of common base current gain versus frequency
for any transistor has these characteristics, and the same
general appearance as the curve of Figure 1.
The common emitter parameter which corresponds to fαb,
is fαe, the common emitter current gain cutoff frequency.
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/α+ 0/+ 0/+6
- $"$% &% 2>
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2> 2>
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Figure 1. Graph Represents a Curve of
a Common Base Current Gain Plotted
Against Frequency Variations
* Circuit diagrams are included as a means of illustrating typical semiconductor applications, consequently, complete information sufficient for
construction purposes is not necessarily given. The information in this application note has been carefully checked, and is believed to be entirely
reliable. However, no responsibility is assumed for inaccuracies. Furthermore, such information does not convey to the purchaser of the
semiconductor devices described any license under the patent rights of Motorola Inc. or others.
Reprinted from ELECTRONIC PRODUCTS
REV 0
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GLOSSARY OF SYMBOLS
Symbol
Definition
hfb
Common base ac forward current gain (alpha)
hfbo
Value of hfb at 1 kHz
hfe
Common emitter ac forward current gain
(beta)
hfeo
Value of hfe at 1 kHz
fαb
Common base current gain cutoff frequency.
Frequency at which hfb has decreased to a
value 3 dB below hfbo. (hfb = 0.707 hfbo)
Common emitter current gain cutoff frequency. Frequency at which hfe has decreased to a
value of 3 dB below hfeo (hfe = 0.707 hfeo)
Gain bandwidth product. Frequency at which
hfe = 1 (0 dB)
fT
Gpe
Common emitter power gain
fmax
Maximum frequency of oscillation. Frequency
at which Gpe = 1 (0 dB)
Kθ
Excess phase shifter factor. Factor which is a
function of excess phase shift of current in the
base of a transistor.
By definition, fαe is the frequency at which hfe (beta, the
common emitter of ac short–circuit current gain), has
decreased 3 dB below hfeo (the value of hfe at 1 kHz). A
typical curve of hfe versus frequency for a transistor with an
fαe of 100 kHz is shown in Figure 2.
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Figure 2. Common Emitter Current Gain is Plotted
Against Frequency in the Curve Shown
This curve also has the significant characteristics listed
for Figure 1. These characteristics allow such a curve to be
constructed for a particular transistor by knowing only hfeo
and fαe. From the curve, hfe at any frequency could be
determined. Furthermore, if fαe, is not known, a curve could
also be constructed if hfeo and hfe at any frequency above
fαe were known. Thus to determine hfe at any frequency, it
is necessary to know only hfeo and either fαe or hfe at some
frequency f, where f is greater than fαe.
Sometimes hfeo is needed and only hfbo is given, or vice
versa. The quantities hfbo and hfeo are related by the
following:
2
hfbo
1 – hfbo
(1)
hfbo =
hfeo
hfeo + 1
(2)
Equations 1 and 2 are plotted in Figure 3. To further
facilitate computations, the low frequency current gain scales
of Figures 7–8 contain both an hfbo and an hfeo scale, and
may be entered with a knowledge of either quantity.
0/+6
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fαe
hfeo =
0/.6 β6
Figure 3. The Relationship Between hfeo and
hfbo is given by the Graph Shown
RELATIONSHIPS BETWEEN fαe AND fαb
Suppose two transistors are considered for a particular
application where performance at high frequencies is of
interest. The data sheets are compared and it is discovered
that one specifies fαb and the other fαe. What preliminary
comparisons can be made from this without making any
laboratory measurements?
Phillips1 gives a discussion of the relationships between
fαe and fαb with the following result:
fαe = Kθ (1 – hfbo) fαb
(3)
where Kθ is a function of excess phase shift in the base
region and has some value between 0.5 and 1.0.
Most transistors have a Kθ in the 0.8 to 1.0 range. Alloy
transistors have a Kθ of 0.82.
The nomograms provide solutions for values of Kθ of 0.9
and 0.8. If more specific information on Kθ is not available,
a value of 0.8 is recommended.
The quantity fαe is normally a much lower frequency than
fαb for the same transistor. For example, consider the
Motorola 2N1141 germanium transistor. The data sheets give
typical values fαb = 1,000 MHz and hfbo = 0.98. Substituting
in Equation 3 yields fαe = 0.80 (1 – 0.98) 1,000 = 16 MHz.
This result is in approximate agreement with the hfe
versus frequency curve of the manufacturer’s 2N1141 data
sheet.
For the practical application of Equation 3, refer to Figure
7. When any two of the quantities fαe, fαb, hfbo, or hfeo are
known, use the nomograms to find the third quantity.
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Table 1. Conversion Table for Power, Voltage, and
Current Ratios into Decibels
dB
Power
Ratio
Voltage
or
Current
Ratio
0
0.5
1.0
1.5
2.0
3.0
4.0
5.0
6.0
7.0
8.0
9.0
1.00
1.12
1.26
1.41
1.58
2.00
2.51
3.16
3.98
5.01
6.31
7.94
1.00
1.06
1.12
1.19
1.26
1.41
1.58
1.78
2.00
2.24
2.51
2.82
dB
Power
Ratio
Voltage
or
Current
Ratio
10
15
20
25
30
40
50
60
10.0
31.5
100
316
1,000
10,000
105
106
3.2
5.6
10
18
32
100
316
1,000
A common high–frequency parameter is fT, the gain
bandwidth product and is defined as that frequency at which
hfe = 1 (0 dB).
The value fT is sometimes specified indirectly on
high–frequency transistor data sheets. This is done by
specifying hfe at some frequency above fαe, thus fT is then
obtained by multiplying the magnitude of hfe by the frequency
of measurement. This relationship arises from the 6 dB per
octave characteristic of the hfe versus frequency curve above
fαe. Since 6 dB represents a current gain magnitude of 2,
hfe is halved each time frequency is doubled, and vice versa.
Therefore, the product of hfe and frequency of the sloping
portion of the curve yields fT.
For example, consider the Motorola 2N2218 silicon
annular transistor. The data sheet gives a typical hfe of 4.0
at 100 MHz. Multiplication of hfe times the frequency of
measurement yields fT = 4.0 x 100 = 400 MHz. This is in
agreement with the data sheet which specifies a typical fT
of 400 MHz.
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/& 0/.6 /α.
0/.6 2> 0/.
/&
/
/α.
0/.6
/&
/& 0/. /
/ /α.
Figure 4. This Nomogram is Useful in Finding hfe,
when a Frequency f > fαe
Figure 5. The Quantity fT is Found from this
Nomogram Once fαe and hfeo are Known
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/4*<
/4*< / √
/
/α.
) &! /α. %%
& % θ Figure 6. Maximum Frequency is Found from this
Nomogram Knowing the Frequency and Power Gain
The parameter fT is also equal to the product of hfeo and
fαe, expressed by
fT = hfeo x fαe
(4)
with hfeo known, Equation 4 provides a simple means of
finding fαe when fT is known or vice versa. (See Figure 5.)
Phillips also develops the following relationship between
fαb and fT:
fT = Kθ hfbo fαb
(5)
where Kθ is the same quantity as in Equation 3. Notice that
since Kθ lies between 0.5 and 1.0, the fT of a transistor is
approximately equal to or slightly less than its fαb. (See
Figure 8.)
RULES FOR DETERMINING hfe
The following rules summarize how to determine hfe at
some frequency f:
Rule 1: When f < fαe, hfe ≈ hfeo
Rule 2: When f fαe, hfe ≈ 0.7 hfeo
Rule 3: When f > fαe, consider hfe to be decreasing at
6 dB per octave at frequency f and use Figure 4
to find hfe.
Rule 4: (A) If hfbo not hfeo is known, use Figure 3 to find
hfeo.
4
0/.6 0/+6
/α+
/α. θ 0/+6/α+
0/+6 2> 0/+ 0/.6 2> 0/.
$& % θ Figure 7. Once fαb is Known this Nomogram
is Used to Find fαe
(B) If hfeo and fαe are known, use Figure 5 to find
fT. Use Figure 8 to find fT if fαb is known.
(C) If fT is known, use Figure 5 to find fαe. Use
Figure 7 to find fαe if fαb is known.
Though common emitter current gain is equal to 1 at fT,
there may still be considerable power gain at fT due to
different input and output impedance levels. Thus, fT is not
necessarily the highest useful frequency of operation of a
transistor, and an additional parameter, the maximum
frequency of oscillation (fmax), is sometimes encountered.
The term fmax is the frequency at which common emitter
power gain is equal to 1, and is related to fT by
fmax fT
8π r′bCc
(6)
where r′b is the base resistance and Cc is the collector
capacitance.
A plot of common emitter power gain versus frequency
also has the characteristics shown in Figure 1. This leads
to another gain bandwidth product
fmax f
Power Gain
(7)
where f is the frequency of measurement and power gain
is expressed in magnitude not in decibels. Hence, f max
may be found by measuring power gain at some frequency
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on the 6 dB per octave portion of the power gain versus
frequency curve, and multiplying the square root of the
power gain with the frequency of measurement (see
Figure 6). The symbol for common emitter power gain is Gpe.
The parameters are voltage and current dependent, and
operating point must be considered in all cases. For
example, the high–frequency hfe measurement at one
collector voltage and current must not be used to calculate
fT directly at another voltage and/or current without
considering the added effects of the different operating point.
The parameter fαe for present high frequency transistors
usually lies in the region between 100 and 500 MHz. The
term hfe, measured at any frequency above this region is
assumed on the 6 dB per octave portion of hfe versus
frequency curve and is used to calculate fT directly.
Power gain measured at any frequency above 500 MHz
is assumed on the 6 dB per octave portion of the power gain
versus frequency curve and is used to calculate f max
directly.
/α+
/&
∞
0/+6
To find hfeo when hfbo is known or vice versa, enter Figure
3 with the known value and read the unknown directly. Given:
hfbo = 0.96. Find: hfeo. Answer = 24.
EXAMPLE 2
There are no special instructions for the nomogram of
Figure 5, merely use it to find the unknown parameter when
any two are known. Given: hfeo = 40 and fT = 400 MHz. Find:
fαe. Answer: 10 MHz.
EXAMPLE 4
Figure 6 is a nomogram of f max and common emitter
power gain measured at some frequency f where power gain
is known to be decreasing at 6 dB per octave. Given: power
gain at 500 MHz is 6 dB. Find f max. Answer: 800 MHz.
Given: f max = 1000 MHz. Find: power gain at 250 MHz.
Answer: 12 dB.
EXAMPLE 5
) &! /& %%
& % θ EXAMPLE 1
EXAMPLE 3
0/.6
contain two frequency scales. Decimal points may be shifted
on the frequency scales of any nomogram as long as they
are shifted the same amount on both scales (i.e., both
frequency scales of a nomogram must be multiplied by 10
to the same power). This enables the same nomogram to
be used for both high and low–frequency transistors.
The nomograms assume that both power gain and current
gain decrease with increasing frequency at a rate of 6 dB
per octave at high frequencies.
All power gain and current gain scales (except hfbo and
hfeo) are calibrated in both actual magnitudes and decibel
values for convenience.
Figure 4 is a nomogram of fT and hfe at some frequency
f, where f > fαe. Given: hfe at 100 MHz is 6 dB. Find: hfe
at 75 MHz. Answer: 4, or 12 dB.
/& θ < 0/+6 < /α+
0/+6 2> 0/+
0/.6 2> 0/.
$& % θ Figure 8. This Nomogram Represents fT, fαb, and
Either hfbo or hfeo
INSTRUCTIONS FOR CURVES AND NOMOGRAMS
The nomograms assume no shift in operating point.
Known parameters used to find an unknown must be
measured at the same collector voltage and collector current
as the desired unknown.
Frequency scales on the nomograms are calibrated in
numbers only without units. Furthermore, all nomograms
Figure 7 is a nomogram of fαb , fαe, and either h fbo or
hfeo. To account for variations in this relationship with
different transistor types, there are two fαe scales, one for
Kθ = 0.8 and one for Kθ = 0.9. Given: fαe = 1 MHz and hfbo =
0.90. Find: fαe. Answer: 80 kHz (assuming Kθ = 0.8).
EXAMPLE 6
Figure 8 is a nomogram of fT, fαb , and either h fbo or h feo.
To account for variations in this relationship with different
transistor types, there are two f T scales, one for Kθ = 0.8
and one for Kθ = 0.9. Given: f T = 400 MHz and hfbo = 0.90.
Find: fαb. Answer: 555 MHz (assuming Kθ = 0.8).
REFERENCE
1. A. B. Phillips, “Transistor Engineering”, McGraw–Hill
Book Company, Inc., New York, N.Y., Chapter 14.
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NOTES
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NOTES
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